Final answer:
In the given sentence, 'does' fills the first blank to agree with the singular subject 'Lamar', and 'someone' or another noun/pronoun fills the second blank to indicate who hired Lamar.
Step-by-step explanation:
The sentence should be completed as follows:
Lamar does have a duty to win the race because someone hired Lamar to drive his car in the race.
The blank before "have a duty" should be filled with a helping verb that is compatible with the subject "Lamar". In this case, the correct helping verb is "does" because it agrees with the singular subject. The second blank should be filled with a noun or pronoun that indicates who hired Lamar, which could be a specific name or a pronoun such as "someone", "a team", or "they".